Question

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros:$-2,2,3;$ degree $3$

Answer 1

Express the polynomial functions whose real zeros and degree are given.

Given data,

Zeros:$-2,2,3;$ degree $3$

Standard form of polynomial function can be expressed as,

$f\left(x\right)=a\left(x-r_1\right)\left(x-r_2\right)\left(x-r_3\right)$

Here, $a$ can be any real number and $r_1,r_2,\text{and}{r}_3$ are the zeros of the polynomial.

From the given data we can rewrite the polynomial function as,

$$\begin{gathered} f\left(x\right)=a\left(x-r_1\right)\left(x-r_2\right)\left(x-r_3\right) \\ \Rightarrow f\left(x\right)=1\left\{x-\left(-2\right)\right\}\left(x-2\right)\left(x-3\right) \\ \Rightarrow f\left(x\right)=\left(x+2\right)\left(x-2\right)\left(x-3\right) \\ \Rightarrow f\left(x\right)=\left(x^2-4\right)\left(x-3\right) \\ \Rightarrow f\left(x\right)=x^3-4x-3x^2+12 \\ \Rightarrow f\left(x\right)=x^3-3x^2-4x+12 \end{gathered}$$

Therefore, the polynomial function is $f\left(x\right)=x^3-3x^2-4x+12$.